Theory of the many-body localization transition in one dimensional systems

TL;DR

This paper develops a real space renormalization group theory for the many-body localization transition in one-dimensional systems, predicting a Griffiths phase with sub-diffusive transport and a new infinite randomness fixed point controlling the transition.

Contribution

It introduces a novel RG approach to describe the MBL transition, including a phenomenological model and predictions about the critical behavior and Griffiths phase properties.

Findings

The delocalized phase near the transition is a Griffiths phase with sub-diffusive transport.

The transition is governed by a new infinite randomness RG fixed point.

Entanglement entropy exhibits logarithmic growth at the critical point.

Abstract

We formulate a theory of the many-body localization transition based on a novel real space renormalization group (RG) approach. The results of this theory are corroborated and intuitively explained with a phenomenological effective description of the critical point and of the "badly conducting" state found near the critical point on the delocalized side. The theory leads to the following sharp predictions: (i) The delocalized state established near the transition is a Griffiths phase, which exhibits sub-diffusive transport of conserved quantities and sub-ballistic spreading of entanglement. The anomalous diffusion exponent $\alpha < 1/2$ vanishes continuously at the critical point. The system does thermalize in this Griffiths phase. (ii) The many-body localization transition is controlled by a new kind of infinite randomness RG fixed point, where the broadly distributed scaling variable is closely related to the eigenstate entanglement entropy. Dynamically, the entanglement grows as $\sim\log t$ at the critical point, as it also does in the localized phase. (iii) In the vicinity of the critical point the ratio of the entanglement entropy to the thermal entropy, and its variance (and in fact all moments) are scaling functions of $L/\xi$, where $L$ is the length of the system and $\xi$ is the correlation length, which has a power-law divergence at the critical point.